UPDATE: In the comments to that piece, reader J.T. Wenting observes:
Message: Space elevators most likely will be built from space down towards earth rather than from the surface up.
Would they still be an extension of the country they're anchored to or would they be space structures reaching the surface?
I'd say the latter, similar to a ship mooring in a harbour not being real estate of the country that harbour is located in, as technically the space elevator would be moored to the ground rather than being built on it.
The problems associated with anchoring such a beast in an unstable and/or corrupt equatorial country has caused many of those planning such things to put them instead on floating ocean platforms, in international waters. This raises some new issues, because now, instead of (as Glenn notes) the structure simply being a very high tower, it would now be a tall ship that would put to shame all of the previous false claimants to that designation, with their puny little sticks for masts.